2. The discharge through a Francis turbine rotating at 150 rpm is 1.5 m3s-1. The outer and inner diameters of the turbine runner are 1500 mm and 1000 mm respectively. The runner thicknesses at entry and exit are 100 mm and 250 mm respectively. The power transfer to the runner is 150 kW. There is no shock at the inlet and outlet of the runner. Assume no whirl at outlet.
a) Find the whirl velocity at inlet.
b) What is the angle of the guide vanes?
c) Find the ideal entry and exit blade angles assuming no shock.
a) To find the whirl velocity at the inlet, we need to calculate the tangential velocity of the runner at the inlet.
First, we find the radius at the inlet:
radius_inlet = (outer diameter / 2) - runner thickness_at_entry
= (1500 mm / 2) - 100 mm
= 650 mm
Next, we convert the tangential speed from rpm to m/s:
tangential_speed_inlet = 2π * radius_inlet * (150 rpm / 60)
= 2π * 0.65 m * (150 / 60)
≈ 13.09 m/s
Finally, we find the whirl velocity at the inlet by subtracting the axial velocity from the tangential velocity at the inlet:
whirl_velocity_inlet = tangential_speed_inlet - axial_velocity_inlet
b) To find the angle of the guide vanes, we need to use the concept of absolute velocity triangle and the law of conservation of angular momentum.
The absolute velocity triangle consists of the absolute velocity at the inlet, the whirl velocity, and the relative velocity. The angle between the absolute velocity and the relative velocity is the blade angle.
Since there is no whirl at the outlet and no shock, the whirl velocity at the outlet is zero. Therefore, we can assume that the whirl velocity at the inlet is also zero.
As a result, the absolute velocity at the inlet is equal to the relative velocity at the inlet, which is the axial velocity at the inlet.
Therefore, the angle of the guide vanes is equal to the angle of the axial velocity at the inlet.
c) To find the ideal entry and exit blade angles assuming no shock, we need to use the concept of the relative velocity triangle.
The relative velocity triangle consists of the relative velocity at the inlet, the relative velocity at the outlet, and the blade angles at the entry and exit.
To find the ideal entry and exit blade angles, we need to know the values of the tangential velocities at the inlet and outlet.
First, we find the radius at the exit:
radius_exit = (outer diameter / 2) - runner thickness_at_exit
= (1500 mm / 2) - 250 mm
= 500 mm
Next, we convert the tangential speed from rpm to m/s:
tangential_speed_exit = 2π * radius_exit * (150 rpm / 60)
= 2π * 0.5 m * (150 / 60)
≈ 15.71 m/s
Finally, we find the ideal entry and exit blade angles using the relative velocity triangles. These calculations involve trigonometry and specific geometry values of the runner, which are not provided in the question.